Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 1, pp. 203-214, Warsaw 2013

PHENOMENOLOGICAL MODELING OF MECHANICAL PROPERTIES

OF METAL FOAM

Tadeusz Wegner, Dariusz Kurpisz

Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland

e-mail: darek kurpisz@o2.pl; dariusz.kurpisz@put.poznan.pl

In this paper, constitutive relations formetal foamunder complex load are introduced using
a phenomenological concept.On the basis of stress-strainand transversal-longitudinal strain
dependencies in uniaxial tests, a form of the function for strain energy density is specified,
and further, the criterion for loss of stability due to the occurrence of plastic flow is defined.
The theoretical investigations are illustrated by a short numerical example.

Key words:metal foam, strain energy density function, phenomenological method of mode-
ling, energy conservation principle

1. Introduction

Modeling of mechanical properties can give an answer to the question about conditions of ma-
terial damage. This phenomenon was taken into account by Petryk (1991) and Perzyna (2005).
We have here a fundamental description of material properties with the use of displacement,
strain and stress tensors for continuous materials. Recently, metal foams are materials which
have gained wide use. Inmany papers, attention was focused on good properties of foams, such
as: lowweight, energy absorption and non-flammability. Based on experimental results, amodel
of the porous material was investigated by Gibson and Ashby (1999). The impact of technolo-
gical process of manufacturing or microstructural parameters on varied properties of the foam
was investigated by Koza et al. (2002). A large number of scientific and research works are
devoted to engineering employment of foams. Hutchinson andHe (2000) took into account buc-
kling of cylindrical sandwich shells with metal foam cores. Mahjoob and Vafai (2008) discussed
an influence of microstructural parameters of a metal foam such as porosity and density on
heat transfer. Unfortunately, most of available publications do not provide information about
physical mechanisms of metal foams destruction. For this reason, an energymethod based on a
phenomenological approach is promising, which involves construction of a strain energy density
function only on the basis of relations between stress and strain in longitudinal and transversal
directions, derived from the uniaxial tension test of thematerial sample, without any knowledge
of the internal structure of the material. The material is interpreted here as a black box. The
input signal is a loading program, and the output signal is the deformation state. The identifi-
cation of amanner of processing of this signal is equivalent tomodel construction of mechanical
properties of material.

Application of the energy conservation principle to thedescription and identification of stable
or unstable states of aluminum was presented by Wegner and Kurpisz (2009). In the paper
presented now, the essential problem is the use of experimental characteristics from uniaxial
tension tests to construct the strain energy function for an aluminium foam.



204 T.Wegner, D. Kurpisz

2. Uniaxial tension tests and their approximations for the aluminium foam

To obtain basic information about physical properties of the aluminium foam, uniaxial tension
and compression tests were made (Grant No. 0807/B/T02/2010/38 supported by the Ministry
of Science and Higher Education in Poland). In the beginning, a cubic sample with length of
the edge 50mmwas subjected to a static uniaxial tension test and then a compression test with
simultaneous measurement of the longitudinal force and longitudinal and transversal strains.
The plots obtained in the tension test are presented in Fig. 1.

Fig. 1. Experimental relation between stress and strain for uniaxial tension (black line) and its
approximation (Eq. (2.1), gray line)

The approximation of experimental results, σA(εx) [MPa], can be expressed as

σA(εx)=

{
−1963261ε2x +1999εx for 0¬ εx ¬ 0.00038
414(εx−0.00038)+0.476 for 0.00038<εx < 0.001

(2.1)

Hence, the longitudinal stiffness coefficient, ẼA(εx) [MPa], takes the form

ẼA(εx)=
σA(εx)

εx
=





−1963261εx +1999 for 0¬ εx ¬ 0.00038

414+
0.31868

εx
for 0.00038<εx < 0.001

(2.2)

Fig. 2. Stiffness coefficient (black line) and its approximation (gray line)

The characteristics obtained for the uniaxial compression test with cyclic unloading are
presented in Figs. 3-5.



Phenomenological modeling of mechanical properties of metal foam 205

Fig. 3. Experimental relation between stress and strain (black line) and its approximation (gray line) in
the compression test with cyclic unloading

Fig. 4. Stiffness coefficient (black line) and its approximation (gray line) in the compression test with
cyclic unloading

Fig. 5. Experimental relation between transversal and longitudinal strain (black line) and its
approximation (gray line) in the compression test with cyclic unloading

The approximation of stress-strain (σA(εx) [MPa]) relation for uniaxial compression can be
written in the form

σA(εx)=






426εx for −0.00137¬ εx ¬ 0
−
√
−306εx−0.079 for −0.0088¬ εx ¬−0.00137

47.3(εx+0.0088)−1.62 for −0.015<εx < 0.0088
(2.3)



206 T.Wegner, D. Kurpisz

Hence, the approximation of the stiffness coefficient, ẼA(εx) [MPa], can be expressed as

ẼA(εx)=





426 for −0.00137¬ εx ¬ 0
√
−306εx−0.079
−εx

for −0.0088¬ εx ¬−0.00137

47.3
(
1+
0.0088

εx

)
− 1.62
εx

for −0.015<εx < 0.0088

(2.4)

The transversal strain can be approximated by the function

εy(εx)=−5.3ε2x−0.48εx where −0.018¬ εx ¬ 0 (2.5)

or using the coefficient of transversal strain

ν̃(εx)=−
εy

εx
=5.3εx+0.48 where −0.018¬ εx ¬ 0 (2.6)

3. Geometrical interpretation of the deformation process

The deformation process of ametal foam can be presented by the use of geometrical interpreta-
tion. Because the foam can be regarded as an isotropic material (for themean distance of pores
considerably smallest than the dimension of a material piece and the same in all directions),
the coefficients of stiffness and transversal strain are the same in different directions. However,
the coefficients of stiffness and transversal strain do not have constant values, like the elastic
modulus andPoisson ratio inmost linear elastic and isotropic material models described in lite-
rature. That is why the relations between strain and stress components undergo complication.
Let us take the space of main components of the deformation state. Each deformation process
is represented by the curve whose points are determined by the current deformation state.
Every displacement along the deformation path C needs a work. Because the (ε1×ε2×ε3)-

space is potential, thework is not dependent on the shape of the path C. The relations between
the strain and stress components take the form

ε1 =
σ1

E(ε1)
−ν(ε2)

σ2

E(ε2)
−ν(ε3)

σ3

E(ε3)

ε2 =−ν(ε1)
σ1

E(ε1)
+
σ2

E(ε2)
−ν(ε3)

σ3

E(ε3)

ε3 =−ν(ε1)
σ1

E(ε1)
−ν(ε2)

σ2

E(ε2)
+
σ3

E(ε3)

(3.1)

and after a series of transformations, we have the following relations

σi(ε1,ε2,ε3)=E(εi)
Ai(ε1,ε2,ε3)

Bi(ε1,ε2,ε3)
i=1,2,3 (3.2)

where
A1(ε1,ε2,ε3)= ν(ε2)[1+ν(ε3)](ε1−ε2)− [1+ν(ε2)][ε1+ε3ν(ε3)]
B1(ε1,ε2,ε3)= ν(ε2)[1+ν(ε1)][1+ν(ε3)]− [1+ν(ε2)][1−ν(ε1)ν(ε3)]
A2(ε1,ε2,ε3)= ν(ε3)[1+ν(ε1)](ε2−ε3)− [1+ν(ε3)][ε2+ε1ν(ε2)]
B2(ε1,ε2,ε3)= ν(ε3)[1+ν(ε2)][1+ν(ε1)]− [1+ν(ε3)][1−ν(ε1)ν(ε2)]
A3(ε1,ε2,ε3)= ν(ε1)[1+ν(ε2)](ε3−ε1)− [1+ν(ε1)][ε3+ε2ν(ε2)]
B3(ε1,ε2,ε3)= ν(ε1)[1+ν(ε3)][1+ν(ε2)]− [1+ν(ε1)][1−ν(ε2)ν(ε3)]

(3.3)

and ν(εi) is the approximation of the transversal strain coefficient.



Phenomenological modeling of mechanical properties of metal foam 207

Let us assume that the deformation path C can be written as

C :





ε1 = ε
K
1 t

ε2 = ε
K
2 t

ε3 = ε
K
3 t

(3.4)

where t ∈ 〈0,1〉 and εKi , i= 1,2,3 are the end components of the deformation state. Putting
(3.4) to (3.2) and (3.3), we receive

σi(ε
K
1 ,ε

K
2 ,ε

K
3 , t)=E(ε

K
i t)
Ai(ε

K
1 ,ε

K
2 ,ε

K
3 , t)

Bi(ε
K
1 ,ε

K
2 ,ε

K
3 , t)

i=1,2,3 (3.5)

where

A1(ε
K
1 ,ε

K
2 ,ε

K
3 , t)= ν(ε

K
2 t)[1+ν(ε

K
3 t)](ε

K
1 t−εK2 t)− [1+ν(εK2 t)][εK1 t+εK3 tν(εK3 t)]

B1(ε
K
1 ,ε

K
2 ,ε

K
3 , t)= ν(ε

K
2 t)[1+ν(ε

K
1 t)][1+ν(ε

K
3 t)]− [1+ν(εK2 t)][1−ν(εK1 t)ν(εK3 t)]

A2(ε
K
1 ,ε

K
2 ,ε

K
3 , t)= ν(ε

K
3 t)[1+ν(ε

K
1 t)](ε

K
2 t−εK3 t)− [1+ν(εK3 t)][εK2 t+εK1 tν(εK2 t)]

B2(ε
K
1 ,ε

K
2 ,ε

K
3 , t)= ν(ε

K
3 t)[1+ν(ε

K
2 t)][1+ν(ε

K
1 t)]− [1+ν(εK3 t)][1−ν(εK1 t)ν(εK2 t)]

A3(ε
K
1 ,ε

K
2 ,ε

K
3 , t)= ν(ε

K
1 t)[1+ν(ε

K
2 t)](ε

K
3 t−εK1 t)− [1+ν(εK1 t)][εK3 t+εK2 tν(εK2 t)]

B3(ε
K
1 ,ε

K
2 ,ε

K
3 , t)= ν(ε

K
1 t)[1+ν(ε

K
3 t)][1+ν(ε

K
2 t)]− [1+ν(εK1 t)][1−ν(εK2 t)ν(εK3 t)]

(3.6)

Expressions (3.5) assign the stress state (σ1,σ2,σ3) to an optional indirect deformation state
(εK1 t,ε

K
2 t,ε

K
3 t).

4. Strain energy density function

Let us assume that we want to appoint the work of change of the deformation state from P1
to P2 along path C (see Fig. 6). Because the discussed points are very close to each other,

Fig. 6. A concept of the material deformation process due to its volumetric part

so the generalized loads, realizing this slip, assume a constant value. If we accept that current
dimensions of the elementary cube in the deformation state P1 are l1× l2× l3, and respectively
in P2 – (l1+∆l1)× (l2+∆l2)× (l3+∆l3), then the increment of work can be written as

∆WP1P2 =F1∆l1+F2∆l2+F3∆l3 (4.1)

After dividing (4.1) by the volume in the initial state V0, we have

∆LP1P2

V0
=
F1

l02l
0
3

∆l1

l01
+
F2

l01l
0
3

∆l2

l02
+
F3

l01l
0
2

∆l3

l03
=
3∑

i=1

σi
∆li

l0i
(4.2)



208 T.Wegner, D. Kurpisz

Hence, the increment of work density caused by the increment of deformation state compo-
nents ∆ε= [∆ε1,∆ε2,∆ε3] takes the form

∆LP1P2

V0
=
3∑

i=1

σi∆εi (4.3)

Hence, on the base of definition of the contour integral, we can write

∆L

V0
=

∫

C

σ1 dε1+

∫

C

σ2 dε2+

∫

C

σ3 dε3 (4.4)

and finally, by using the energy conservation principle, we have

W =

1∫

0

( 3∑

i=1

σi(ε
K
1 ,ε

K
2 ,ε

K
3 , t)ε

K
i

)
dt= f̂(εK1 ,ε

K
2 ,ε

K
3 ) (4.5)

where W is the strain energy density function. The analytical form of W can be indicated by
substituting (3.5) to (4.5).

4.1. Extraction of volumetric part of energy

Apure volume deformation process takes place if thematerial is under hydrostatic pressure.
In this case, (3.1) takes the form

εV1 =
ks

E(εV1 )
−ν(εV2 )

ks

E(εV2 )
−ν(εV3 )

ks

E(εV3 )

εV2 =−ν(εV1 )
ks

E(εV1 )
+
ks

E(εV2 )
−ν(εV3 )

ks

E(εV3 )

εV3 =−ν(εV1 )
ks

E(εV1 )
−ν(εV2 )

ks

E(εV2 )
+
ks

E(εV3 )

(4.6)

where s ∈ 〈0,1〉 is dimesionless quantity and k < 0. The solution to the system of equations
(4.6) is parametric and gives a pure volumetric deformations path

CV :





εV1 =ϕ1(s)

εV2 =ϕ2(s)

εV3 =ϕ3(s)

(4.7)

so the pure volumetric deformation energy can be written in the form

WV =
3∑

i=1

∫

CV

σVi dε
V
i =

∫

CV

( 3∑

i=1

ksϕ′i(s)
)
ds (4.8)

Received relation (4.8) can not be directly used to extract the volumetric part of energy. It
is necessary to find a relation between the deformation states P1 and P

′

1 (see Fig. 6). On the
base of (3.4) and (4.7), we have

∆V

V
=
3∏

i=1

[1+εi(t)]−1= a(t)−1 (4.9)



Phenomenological modeling of mechanical properties of metal foam 209

and on the other hand

∆V

V
=
3∏

i=1

[1+ϕi(s)]−1= b(s)−1 (4.10)

hence

3∏

i=1

[1+ϕi(s)] =
3∏

i=1

(1+εKi t) (4.11)

and

s=h(εK1 ,ε
K
2 ,ε

K
3 , t)= b

−1(a(εK1 ,ε
K
2 ,ε

K
3 , t)) (4.12)

On the base of (4.12), the volumetric part of energy can be written as

WV =
3∑

i=1

∫

CV

σi dε
V
i

=

1∫

0

[ 3∑

i=1

σi(ε
K
1 ,ε

K
2 ,ε

K
3 , t)ϕ

′

i(h(ε
K
1 ,ε

K
2 ,ε

K
3 , t))

]
h′(εK1 ,ε

K
2 ,ε

K
3 , t) dt

(4.13)

so, it is a function of the deformation state ε=(εK1 ,ε
K
2 ,ε

K
3 ).

5. Stability assumptions

Based on the Jaunzemis criterion of material stability (1967) and the strength hypothesis pre-
sented by Wegner (2000a,b, 2005, 2009), the damage of material begins in the place, where, in
geometrical interpretation, the state of loss of convexity of the strain energy density function
appears.

The stability regions are there where the strain energy density function is convex. So, the
stability criterion takes the form

δ2W =
3∑

i=1

3∑

j=3

∂2W

∂εi∂εj
δεi δεj > 0 (5.1)

and on the base of the Sylvester theorem about quadratic form, we have

detM> 0

detM1 > 0 detM2 > 0 detM3 > 0

detM1,2 > 0 detM1,3 > 0 detM2,3 > 0

(5.2)

where

M=




∂2W

∂ε21

∂2W

∂ε1∂ε2

∂2W

∂ε1∂ε3

∂2W

∂ε1∂ε2

∂2W

∂ε22

∂2W

∂ε2∂ε3

∂2W

∂ε1∂ε3

∂2W

∂ε2∂ε3

∂2W

∂ε23






210 T.Wegner, D. Kurpisz

M1 =




∂2W

∂ε22

∂2W

∂ε2∂ε3

∂2W

∂ε2∂ε3

∂2W

∂ε23




M2 =




∂2W

∂ε21

∂2W

∂ε1∂ε3

∂2W

∂ε1∂ε3

∂2W

∂ε23




M3 =




∂2W

∂ε21

∂2W

∂ε1∂ε2

∂2W

∂ε1∂ε2

∂2W

∂ε22




M2,3 =

[
∂2W

∂ε21

]
M3,1 =

[
∂2W

∂ε22

]
M2,3 =

[
∂2W

∂ε23

]

If we take into consideration damage of amaterial due to plastic flow, then the increment of
volume change is equal to zero

δ
(∆V
V

)
= δ((1+ε1)(1+ε2)(1+ε3)−1)= 0

δ
(∆V
V

)
=(1+ε2+ε3)δε1 +(1+ε1+ε3)δε2+(1+ε1+ε2)δε3 =0

(5.3)

and (5.1) can be written in the form

A(ε1,ε2,ε3)+B(ε1,ε2,ε3)[(1+ε2+ε3)δε1+(1+ε1+ε3)δε2+(1+ε1+ε2)δε3]
2

−
[
(1+ε2+ε3)

2B(ε1,ε2,ε3)−
∂2W

∂ε21

]
δε21−

[
(1+ε1+ε3)

2B(ε1,ε2,ε3)−
∂2W

∂ε22

]
δε22

−
[
(1+ε1+ε2)

2B(ε1,ε2,ε3)−
∂2W

∂ε23

]
δε23

−2
[
(1+ε2+ε3)(1+ε1+ε3)B(ε1,ε2,ε3)−

∂2W

∂ε1∂ε2

]
δε1 δε2

−2
[
(1+ε2+ε3)(1+ε1+ε2)B(ε1,ε2,ε3)−

∂2W

∂ε1∂ε3

]
δε1 δε3

−2
[
(1+ε1+ε3)(1+ε1+ε2)B(ε1,ε2,ε3)−

∂2W

∂ε2∂ε3

]
δε2 δε3 =0

(5.4)

where

A(ε1,ε2,ε3)=
∂2W

∂ε21
δε21+

∂2W

∂ε22
δε22+

∂2W

∂ε23
δε23+2

∂2W

∂ε1∂ε2
δε1 δε2+2

∂2W

∂ε1∂ε3
δε1 δε3

+2
∂2W

∂ε2∂ε3
δε2 δε3

B(ε1,ε2,ε3)=
∂2W

∂ε21
+
∂2W

∂ε22
+
∂2W

∂ε23
+2
∂2W

∂ε1∂ε2
+2
∂2W

∂ε1∂ε3
+2
∂2W

∂ε2∂ε3

which, after a series of transformations, gives

δ2W =
[
(1+ε2+ε3)

2B(ε1,ε2,ε3)+
∂2W

∂ε21

]
δε21+

[
(1+ε1+ε3)

2B(ε1,ε2,ε3)+
∂2W

∂ε22

]
δε22

+
[
(1+ε1+ε2)

2B(ε1,ε2,ε3)+
∂2W

∂ε23

]
δε23

+2
[
(1+ε2+ε3)(1+ε1+ε3)B(ε1,ε2,ε3)+

∂2W

∂ε1∂ε2

]
δε1 δε2

+2
[
(1+ε2+ε3)(1+ε1+ε2)B(ε1,ε2,ε3)+

∂2W

∂ε1∂ε3

]
δε1 δε3

+2
[
(1+ε1+ε3)(1+ε1+ε2)B(ε1,ε2,ε3)+

∂2W

∂ε2∂ε3

]
δε2 δε3

(5.5)



Phenomenological modeling of mechanical properties of metal foam 211

Therefore, on the base of the Sylvester theorem, we can write
∣∣∣∣∣∣∣

x11 x12 x13
x21 x22 x23
x31 x32 x33

∣∣∣∣∣∣∣
¬ 0

∣∣∣∣∣
x11 x12
x21 x22

∣∣∣∣∣¬ 0
∣∣∣∣∣
x11 x13
x31 x33

∣∣∣∣∣¬ 0
∣∣∣∣∣
x22 x23
x32 x33

∣∣∣∣∣¬ 0

∣∣∣x11
∣∣∣¬ 0

∣∣∣x22
∣∣∣¬ 0

∣∣∣x33
∣∣∣¬ 0

(5.6)

where for i,j =1,2,3

xii =(1+Θ−εi)2
(∂2W
∂ε21
+
∂2W

∂ε22
+
∂2W

∂ε23
+2
∂2W

∂ε1∂ε2
+2
∂2W

∂ε1∂ε3
+2
∂2W

∂ε2∂ε3

)
+
∂2W

∂ε2i

xij =xji =(1+Θ−εi)(1+Θ−εj)
(∂2W
∂ε21
+
∂2W

∂ε22
+
∂2W

∂ε23
+2
∂2W

∂ε1∂ε2

+2
∂2W

∂ε1∂ε3
+2
∂2W

∂ε2∂ε3

)
+
∂2W

∂εi∂εj

(5.7)

The plastic flow appears if at least one of inequalities (5.6) is satisfied.

6. Example

Let us take that nonlinear porousmaterial characteristics can bewritten in form (2.4) and (2.6).
After a series of transformations according to (3.2)-(3.5), the strain energy density function

(4.5) for three axial pressures takes the form

W(ε1,ε2,ε3)= 426

1∫

0

( 3∑

i=1

εi
Ai(ε1,ε2,ε3, t)

Bi(ε1,ε2,ε3, t)

)
dt (6.1)

and, after a series of transformations, we get

W(ε1,ε2,ε3)=−
328.02(ε21 +ε

2
2+ε

2
3)+604.92(ε1ε2+ε1ε3+ε2ε3)

7.54(ε1 +ε2+ε3)

·
[
1+

0.876

7.54(ε1 +ε2+ε3)
ln
(
1−
7.54(ε1 +ε2+ε3)

0.876

)] (6.2)

The strain energy density function for the subspace Ω= ε1×ε2×0 is presented in Fig. 7.
The solution to equations (4.6) gives the relation

εV1 = ε
V
2 = ε

V
3 =−

0.04s

426−10.6s
(6.3)

Basing on (4.12), the relation between s and t takes the form

s=
426
(
1− 3
√
(1+ε1t)(1+ε2t)(1+ε3t)

)

10.64−10.6 3
√
(1+ε1t)(1+ε2t)(1+ε3t)

(6.4)

and finally, according to (4.13), the volumetric part of energy can be written as

WV (ε1,ε2,ε3)=−41.32(ε1+ε2+ε3)
[
1+

0.876

7.54(ε1 +ε2+ε3)
ln
(
1−
7.54(ε1 +ε2+ε3)

0.876

)]
(6.5)



212 T.Wegner, D. Kurpisz

Fig. 7. Strain energy density function

Fig. 8. Volumetric part of the strain energy density function

The plot of the volumetric part of the strain energy density function is shown in Fig. 9.

Fig. 9. Plot of the strain energy density function and its volumetric part for ε2 = ε3 =0

7. Conclusions

Themain conclusions are:

• Direct application of the known Hook law for description of the relation between stress
and strain inmetal foams is impossible due to nonlinear elastic properties of thematerial.



Phenomenological modeling of mechanical properties of metal foam 213

• The strain energy density function can be used for description of properties of nonlinear
elastic porousmaterials.

• Material characteristics have a significant influence on the shape of plot of the energy
density function and, in consequence, on regions of stability.

• The strain energy density function is symmetric for an isotropic metal foam.
• The volumetric part of the strain energy density function can be properly extracted by as-
suming an appropriate interpretation for finding an adequate equivalent of the hydrostatic
pressure.

• From the point of view of energy, the volumetric deformation is the main form of defor-
mations, and the deformation of the material is stable due to numerical investigations of
inequalities (5.2) for the elastic range εi ∈ 〈−0.001,0〉.

Acknowledgments

This study is supported by the Ministry of Science and Higher Education in Poland – Grant

No. 0807/B/T02/2010/38.

References

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323-324



214 T.Wegner, D. Kurpisz

Modelowanie metodą fenomenologiczną mechanicznych właściwości pian metalowych

Streszczenie

Wniniejszej pracy, bazując na fenomenologicznymsposobiepodejścia,wyprowadzonozwiązki konsty-
tutywne piany metalowej poddanej złożonemu stanowi obciążenia. Na podstawie zależności naprężenia
oraz odkształcenia poprzecznego od odkształcenia podłużnego uzyskanych w jednoosiowych próbach,
wyznaczono wzór opisujący funkcję gęstości energii wewnętrznej, a następnie sformułowano analityczną
postać kryteriów zniszczenia materiału ze względu na wystąpienie plastycznego płynięcia. Rozważania
teoretyczne zilustrowano prostym przykładem numerycznym.

Manuscript received January 2, 2012; accepted for print May 6, 2012